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Properties

Label 2-48-3.2-c6-0-0
Degree $2$
Conductor $48$
Sign $-0.777 + 0.628i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21 + 16.9i)3-s + 169. i·5-s − 2·7-s + (153. − 712. i)9-s − 33.9i·11-s − 2.95e3·13-s + (−2.88e3 − 3.56e3i)15-s − 4.48e3i·17-s − 5.25e3·19-s + (42 − 33.9i)21-s − 1.02e4i·23-s − 1.31e4·25-s + (8.88e3 + 1.75e4i)27-s − 2.20e3i·29-s − 2.28e4·31-s + ⋯
L(s)  = 1  + (−0.777 + 0.628i)3-s + 1.35i·5-s − 0.00583·7-s + (0.209 − 0.977i)9-s − 0.0255i·11-s − 1.34·13-s + (−0.853 − 1.05i)15-s − 0.911i·17-s − 0.766·19-s + (0.00453 − 0.00366i)21-s − 0.842i·23-s − 0.843·25-s + (0.451 + 0.892i)27-s − 0.0904i·29-s − 0.768·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.777 + 0.628i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.777 + 0.628i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0730547 - 0.206630i\)
\(L(\frac12)\) \(\approx\) \(0.0730547 - 0.206630i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (21 - 16.9i)T \)
good5 \( 1 - 169. iT - 1.56e4T^{2} \)
7 \( 1 + 2T + 1.17e5T^{2} \)
11 \( 1 + 33.9iT - 1.77e6T^{2} \)
13 \( 1 + 2.95e3T + 4.82e6T^{2} \)
17 \( 1 + 4.48e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.25e3T + 4.70e7T^{2} \)
23 \( 1 + 1.02e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.20e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.28e4T + 8.87e8T^{2} \)
37 \( 1 - 3.40e4T + 2.56e9T^{2} \)
41 \( 1 - 1.67e4iT - 4.75e9T^{2} \)
43 \( 1 - 6.40e3T + 6.32e9T^{2} \)
47 \( 1 - 1.79e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.92e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.26e5iT - 4.21e10T^{2} \)
61 \( 1 + 6.25e4T + 5.15e10T^{2} \)
67 \( 1 + 4.38e5T + 9.04e10T^{2} \)
71 \( 1 + 6.82e4iT - 1.28e11T^{2} \)
73 \( 1 + 7.30e5T + 1.51e11T^{2} \)
79 \( 1 + 3.40e5T + 2.43e11T^{2} \)
83 \( 1 - 4.96e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.86e5iT - 4.96e11T^{2} \)
97 \( 1 + 2.81e5T + 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.97781675895172901733639878350, −14.37691634881307594006368475625, −12.54812488052901975274724271200, −11.35407272411280269695587433553, −10.49095844248622571703129353979, −9.469413394961391305252114424009, −7.34124134284450123238853323853, −6.23783819092060161334909430956, −4.61319670456226779726741460731, −2.82929026465393684386746883517, 0.10545650039254847364174801472, 1.71985643325594034845848268691, 4.59474599525089693425096431441, 5.72612806215471172906465934690, 7.35746992061375402947811362799, 8.650526912619489569475340799901, 10.13626567086374785707868399141, 11.65222106110824888975741350442, 12.59177331733055821953997339945, 13.23562632078022806002230576974

Graph of the $Z$-function along the critical line